Solution: 2019 Winter Final - 6
Author: Michiel SmidQuestion
What is the coefficient of $x^{20}y^{35}$ in the expansion of $(5x - 3y)^{55}$?
(a)
$-{55 \choose 35} \cdot 5^{20} \cdot 3^{35}$
(b)
${55 \choose 35} \cdot 5^{20} \cdot 3^{35}$
(c)
${55 \choose 20} \cdot 5^{35} \cdot 3^{20}$
(d)
$-{55 \choose 20} \cdot 5^{35} \cdot 3^{20}$
Solution
What is the coefficient of $x^{20}y^{35}$ in the expansion of $(5x - 3y)^{55}$?
$(5x - 3y)^{55}$
$= \sum_{k=0}^{55} \binom{55}{k} (5x)^{k} (-3y)^{55-k}$
$= \binom{55}{20} (5x)^{20} (-3y)^{35}$
$= \binom{55}{20} 5^{20} x^{20} (-3)^{35} y^{35}$
$= - \binom{55}{20} 5^{20} 3^{35} x^{20} y^{35}$
Thus, the coefficient of $x^{20}y^{35}$ is $- \binom{55}{20} 5^{20} 3^{35}$.